论最大阶数为 5 的稀疏图的注入色度指数

Pub Date : 2024-11-07 DOI:10.1007/s10878-024-01234-7

Jian Lu, Zhen-Mu Hong, Zheng-Jiang Xia

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引用次数: 0

摘要

如果对于路径或三角形的任意三条连续边\(e_1, e_2\) 和\(e_3\)，图 G 的 k 边着色（\varphi \）是可注入的，那么\(\varphi (e_1)\ne \varphi (e_3)\)就是可注入的。G 的注入色度指数 \(\chi _i'(G)\)是使 G 允许注入 k 边着色的最小 k。通过放电法，我们证明了任何最大度为 \(\Delta \le 5\) 的图，如果它的最大平均度小于 \(\frac{20}{7}\) (resp.3)，就有\(\chi _i'(G)\le 12\) （resp.13），这改进了 Zhu (2023) 的结果。

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On injective chromatic index of sparse graphs with maximum degree 5

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On injective chromatic index of sparse graphs with maximum degree 5

A k-edge coloring \(\varphi \) of a graph G is injective if \(\varphi (e_1)\ne \varphi (e_3)\) for any three consecutive edges \(e_1, e_2\) and \(e_3\) of a path or a triangle. The injective chromatic index \(\chi _i'(G)\) of G is the smallest k such that G admits an injective k-edge coloring. By discharging method, we demonstrate that any graph with maximum degree \(\Delta \le 5\) has \(\chi _i'(G)\le 12\) (resp. 13) if its maximum average degree is less than \(\frac{20}{7}\) (resp. 3), which improves the results of Zhu (2023).

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